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## I NTEGER PROGRAMMING MODELS

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**Learning Objectives**• Formulate integer programming (IP) models. • Set up and solve IP models using Excel’s Solver. • Understand the difference between general integer and binary integer variables • Understand use of binary integer variables in formulating problems involving fixed (or setup) costs.**Integer Programming Models**• Some business problems can be solved only if variables haveinteger values. • Airline decides on the number of flights to operate in a given sector must be an integer or whole number amount. Other examples: • The number of aircraft purchased this year • The number of machines needed for production • The number of trips made by a sales person • The number of police officers assigned to the night shift.**Some Facts**• Integer variables may be required when the model represents a one time decision (not an ongoing operation). • Integer Linear Programming (ILP) models are much more difficult to solve than Linear Programming (LP) models. • Algorithms that solve integer linear models do not provide valuable sensitivity analysis results.**Types of Integer Variables**• General integer variables and • Binary variables. • General integer variables can take on any non-negative, integer value that satisfies all constraints in the model. • Binary variables can only take on either of two values: 0 or 1.**Types of Integer Programming Problems**• Pure integer programming problems. • All decision variables must have integer solutions. • Mixed integer programming problems. • Some, but not all, decision variables must have integer solutions. • Non-integer variables can have fractional optimal values. • Pure binary (or Zero - One) integer programming problems. • All decision variables are of special type known as binary. • Variables must have solution values of either 0 or 1. • Mixed binary integer programming problems. • Some decision variables are binary, and other decision variables are either general integer or continuous valued.**Models With General Integer Variables**• A model with general integer variables (IP) has objective function and constraints identical to LP models. • No real difference in basic procedure for formulating an IP model and LP model. • Only additional requirement in IP model is one or more of the decision variables have to take on integer values in the optimal solution. • Actual value of this integer variable is limited by the model constraints. (Values such as 0, 1, 2, 3, etc. are perfectly valid for these variables as long as these values satisfy all model constraints.)**Complexities of ILPS**• If an integer model is solved as a simple linear model, at the optimal solution non-integer values may be attained. • Rounding to integer values may result in: • Infeasible solutions • Feasible but not optimal solutions • Optimal solutions.**Some Features of Integer Programming Problems**• Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution • A feasible solution is ensured by rounding down non-integer solution values but may result in a less than optimal (sub-optimal) solution.**Integer Programming Example**Graphical Solution of Maximization Model Maximize Z = $100x1 + $150x2 subject to: 8,000x1 + 4,000x2 $40,000 15x1 + 30x2 200 ft2 x1, x2 0 and integer Optimal Solution: Z = $1,055.56 x1 = 2.22 presses x2 = 5.55 lathes Feasible Solution Space with Integer Solution Points**Why not enumerate all the feasible integer points and select**the best one? • Enumerating all the integer solutions is impractical because of the large number of feasible integer points. • Is rounding ever done? Yes, particularly if: • The values of the positive decision variables are relatively large, and • The values of the objective function coefficients relatively small.**Pure Integer Programming**Example 1: Harrison Electric Company (1 of 8) • Produces two expensive products popular with renovators of historic old homes: • Ornate chandeliers (C) and • Old-fashioned ceiling fans (F). • Two-step production process: • Wiring ( 2 hours per chandelier and 3 hours per ceiling fan). • Final assembly time (6 hours per chandelier and 5 hours per fan).**Pure Integer Programming**Example 1: Harrison Electric Company (2 of 8) • Production capability this period: • 12 hours of wiring time available and • 30 hours of final assembly time available. • Profits: • Chandelier profit $600 / unit and • Fan profit $700 / unit.**Pure Integer Programming**Example 1: Harrison Electric Company (3 of 8) Objective: maximize profit = $600C + $700F subject to 2C + 3F <= 12 (wiring hours) 6C + 5F <= 30 (assembly hours) C, F >= 0 and integer where C = number of chandeliers to be produced F = number of ceiling fans to be produced**Pure Integer Programming**Example 1: Harrison Electric Company (4 of 8) Graphical LP Solution**Pure Integer Programming**Example 1: Harrison Electric Company (5 of 8) • Shaded region 1 shows feasible region for LP problem. • Optimal corner point solution: C = 3.75 chandeliers and F = 1.5 ceiling fans. • Profit of $3,300 during production period. • But, we need to produce and sell integer values of the products. • The table shows all possible integer solutions for this problem.**Pure Integer Programming**Example 1: Harrison Electric Company (6 of 8) Enumeration of all integer solutions**Pure Integer Programming**Example 1: Harrison Electric Company (7 of 8) • Table lists the entire set of integer-valued solutions for problem. • By inspecting the right-hand column, optimal integer solution is: C= 3 chandeliers, F= 2 ceiling fans. • Total profit = $3,200. • The rounded off solution: C = 4 F = 1 Total profit = $3,100.**Pure Integer Programming**Example 2: Boxcar Burger Restaurants (1 of 4) Boxcar Burger is a new chain of fast-food establishments. Boxcar is planning expansion in the downtown and suburban areas. Management would like to determine how many restaurants to open in each area in order to maximize net weekly profit.**Pure Integer Programming**Example 2: Boxcar Burger Restaurants (2 of 4) • Requirements and Restrictions: • No more than 19 managers can be assigned • At least two downtown restaurants are to be opened • Total investment cannot exceed $2.7 million**Pure Integer ProgrammingExample 2: Boxcar Burger Restaurants**(3 of 4) • Decision Variables X1 = Number of suburban boxcar burger restaurants to be opened. X2 = Number of downtown boxcar burger restaurants to be opened. • The mathematical model is formulated next**Pure Integer Programming**Example 2: Boxcar Burger Restaurants (4 of 4) Net weekly profit Total investment cannot exceed $2.7 dollars At least 2 downtown restaurants Not more than 19 managers can be assigned**Pure Integer ProgrammingExample 3: Personnel Scheduling**Problem (1 of 6) • The City of Sunset Beach staffs lifeguards 7 days a week. • Regulations require that city employees work five days. • Insurance requirements mandate 1 lifeguard per 8000 average daily attendance on any given day. • The city wants to employ as few lifeguards as possible.**Pure Integer ProgrammingExample 3: Personnel Scheduling**Problem (2 of 6) • Problem Summary • Schedule lifeguard over 5 consecutive days. • Minimize the total number of lifeguards. • Meet the minimum daily lifeguard requirements • Sun. Mon. Tue Wed. Thr. Fri. Sat. 8 6 5 4 6 7 9 • For each day, at least the minimum required lifeguards must be on duty.**Pure Integer ProgrammingExample 3: Personnel Scheduling**Problem (3 of 6) • Decision Variables: • Xi = the number of lifeguards scheduled to begin on day “I” for i=1, 2, …,7 (i=1 is Sunday) • Objective Function: • Minimize the total number of lifeguards scheduled**Pure Integer ProgrammingExample 3: Personnel Scheduling**Problem (4 of 6) X3 X4 X5 X6 X1 To ensure that enough lifeguards are scheduled for each day, ask which workers are on duty. For example: Who works on Sunday ? Tue. Wed. Thu. Fri. Sun. Repeat this procedure for each day of the week, and build the constraints accordingly.**Pure Integer ProgrammingExample 3: Personnel Scheduling**Problem (5 of 6) • The Mathematical Model**Pure Integer ProgrammingExample 3: Personnel Scheduling**Problem (6 of 6) Note: An alternate optimal solution exists.**General Integer Variable (IP): Mixed Integer Programming**• A mixed integer linear programming model is one in which some, but not all, the variables are restricted integers. • The Shelly Mednick Investment Problem illustrates this situation**Mixed Integer Linear ProgrammingExample 1: Shelly Mednick**Investment Problem (1 of 3) • Shelley Mednick has decided to give the stock market a try. • She will invest in • TCS, a communication company stock, and or, • MFI, a mutual fund. • Shelley is a cautious investor. She sets limits on the level of investments, and a modest goal for gain for the year.**Mixed Integer Linear ProgrammingExample 1: Shelly Mednick**Investment Problem (2 of 3) Data • TCS is been sold now for $55 a share. • TCS is projected to sell for $68 a share in a year. • MFI is predicted to yield 9% annual return. Restrictions • Expected return should be at least $250. • The maximum amount invested in TCS is not to exceed 40 % of the total investment. • The maximum amount invested in TCS is not to exceed $750.**Mixed Integer Linear ProgrammingExample 1: Shelly Mednick**Investment Problem (3 of 3) • Decision variables • X1 = Number of shares of the TCS purchased. • X2 = Amount of money invested in MFI. • The mathematical model Projected yearly return Not more than 40% in TCS Not more than $750 in TCS**Mixed Integer Programming**Example 2: Investment Problem (1 of 2) • $250,000 available for investments providing greatest return after one year. • Data: • Condominium cost $50,000/unit, $9,000 profit if sold after one year. • Land cost $12,000/ acre, $1,500 profit if sold after one year. • Municipal bond cost $8,000/bond, $1,000 profit if sold after one year. • Only 4 condominiums, 15 acres of land, and 20 municipal bonds available.**Mixed Integer Programming**Example 2: Investment Problem (2 of 2) Integer Programming Model: Maximize Z = $9,000x1 + 1,500x2 + 1,000x3 subject to: 50,000x1 + 12,000x2 + 8,000x3 $250,000 x1 4 condominiums x2 15 acres x3 20 bonds x2 0 x1, x3 0 and integer x1 = condominiums purchased x2 = acres of land purchased x3 = bonds purchased**Models With Binary Variables**Binary variables restricted to values of 0 or 1. • Model explicitly specifies that variables are binary. • Typical examples include decisions such as: • Introducing new product (introduce it or not), • Building new facility (build it or not), • Selecting team (select a specific individual or not), and • Investing in projects (invest in a specific project or not).**Any situation that can be modeled by “yes”/“no”,**“good”/“bad” etc., falls into the binary category. • Examples**Pure Binary Integer Programming Models:Example 1: Oil**Portfolio Selection (1 of 7) Firm specializes in recommending oil stock portfolios. • At least two Texas oil firms must be in portfolio. • No more than one investment can be made in foreign oil. • Exactly one of two California oil stocks must be purchased. • If British Petroleum stock is included in portfolio, then Texas-Trans Oil stock must also be included in portfolio. • Client has $3 million available for investments and insists on purchasing large blocks of shares of each company for investment. • Objective is to maximize annual return on investment.**Pure Binary Integer Programming Models:**Example 1. Oil Portfolio Selection (2 of 7) Investment Opportunities**Pure Binary (0, 1) IP Models:**Example 1. Oil Portfolio Selection (3 of 7) Objective: maximize return on investment = $50XT + $80XB + $90XD + $120XH + $110XL + $40XS + $75XC Binary variable defined as: Xi = 1 if large block of shares in company i is purchased = 0 if large block of shares in company iis not purchased where i= T (for Trans-Texas Oil), B (for British Petroleum), D (for Dutch Shell), H (for Houston Drilling), L (for Lonestar Petroleum), S (for San Diego Oil), or C (for California Petro).**Pure Binary IP Models:**Example 1. Oil Portfolio Selection (4 of 7) • Constraint regarding $3 million investment limit expressed as (in thousands of dollars): $480XT + $540XB + $680XD + $1,000XH + $700XL + $510XS + $900XC $3,000 • k Out of n Variables. • Requirement at least two Texas oil firms be in portfolio. • Three (i.e., n = 3) Texas oil firms (XT, XH, and XL) of which at least two (that is, k = 2) must be selected. XT + XH + XL 2**Pure Binary IP Models:**Example 1. Oil Portfolio Selection (5 of 7) • Condition no more than one investment be in foreign oil companies (mutually exclusive constraint). XB + XD 1 • Condition for California oil stock is mutually exclusive variable. • Sign of constraint is an equality rather than inequality. • Simkin mustinclude California oil stock in portfolio. XS + XC = 1**Pure Binary IP Models:**Example 1. Oil Portfolio Selection (6 of 7) • Condition if British Petroleum stock is included in portfolio, then Texas-Trans Oil stock must also be in portfolio. (if-then constraints) XBXT or XB - XT 0 • If XB equals 0, constraint allows XT to equal either 0 or 1. • If XB equals 1, then XT must also equal 1. • If the relationship is two-way (either include both or include neither), rewrite constraint as: XB = XT or XB - XT = 0**Pure Binary IP Models:**Example 1. Oil Portfolio Selection (7 of 7) Objective: maximize return = $50XT + $80XB + $90XD + $120XH + $110XL + $40XS + $75XC subject to $480XT + $540XB + $680XD + $1,000XH + $700XL + $510XS + $900XC $3,000 (Investment limit) XT + XH + XL 2 (Texas) XB + XD 1 (Foreign Oil) XS + XC = 1 (California) XB - XT 0 (Trans-Texas and British Petroleum)**Pure Binary IP Models:**Example 2: Construction Projects (1 of 2) • Recreation facilities selection to maximize daily usage by residents. • Resource constraints: $120,000 budget; 12 acres of land. • Selection constraint: either swimming pool or tennis center (not both). • Data:**Pure Binary IP Models:**Example 2: Construction Projects (2 of 2) Integer Programming Model: Maximize Z = 300x1 + 90x2 + 400x3 + 150x subject to: $35,000x1 + 10,000x2 + 25,000x3 + 90,000x4 $120,000 4x1 + 2x2 + 7x3 + 3x3 12 acres x1 + x2 1 facility x1, x2, x3, x4 = 0 or 1 x1 = construction of a swimming pool x2 = construction of a tennis center x3 = construction of an athletic field x4 = construction of a gymnasium**Pure Binary IP Models:**Example 3: Capital Budgeting (1 of 3) • University bookstore expansion project. • Not enough space available for both a computer department and a clothing department. • Data: